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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<a NAME="E1:12"/><i><font color="Green">E33</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E2:12"/>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D1">ZFMISC_1:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">EX</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i>, </i>;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">EX</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">EX</font> )
 ;<br/><a NAME="E2:12_1"/><i><font color="Green">E36</font></i>: 
( ( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) implies <font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><a NAME="E3:12_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> implies <font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 ;<br/><b>hence </b><a NAME="E4:12_1"/>
<font color="Maroon">A</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">EX</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">EX</font>
 ;<br/><a NAME="E2:12_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  implies <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> )
 ;<br/><a NAME="E3:12_2"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font> implies <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><b>hence </b><a NAME="E4:12_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">EX</font>
 <b>by </b><i>, , <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:12"/>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> ,<font color="Maroon">X</font></span><a href="tarski.html#K2">}</a></span> is   <a href="prob_1.html#NM1">Field_Subset</a> of <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="ref" href="finsub_1.html#D2">FINSUB_1:def 2</a></i>;<br/>


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